Find the equation to the parabola with the focus (a,b) and directrix $\frac{x}{a} + \frac{y}{b} = 1.$

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Solution

Let a point P of coordinate $(x, y)$ be any point on the parabola.
The distance of the point from the focus is $= \sqrt{(x - a)^{2} + (y - b)^{2}}$
The distance of the point from the directrix is $∣ ∣ ∣ \frac{b x + a y - a b}{\sqrt{a^{2} + b^{2}}} ∣ ∣ ∣$

We know that,
$d i s t a n c e o f P f r o m f o c u s = i t s d i s t a n c e f r o m d i r e c t r i x ∴ \sqrt{(x - a)^{2} + (y - b)^{2}} = ∣ ∣ ∣ \frac{b x + a y - a b}{\sqrt{a^{2} + b^{2}}} ∣ ∣ ∣ \Rightarrow (x - a)^{2} + (y - b)^{2} = \frac{b x + a y - a b}{\sqrt{a^{2} + b^{2}}} \Rightarrow (a^{2} + b^{2}) (x^{2} + y^{2} - 2 a x - 2 b y + a^{2} + b^{2}) = b^{2} x^{2} + a^{2} y^{2} + a^{2} b^{2} + 2 a b x y - 2 a^{2} b y - 2 a b^{2} x \Rightarrow a^{2} x^{2} - 2 a b x y + b^{2} y^{2} - 2 a^{3} x - 2 b^{3} y + (a^{4} + a^{2} b^{2} + b^{4}) = 0$

Therefore, this is the equation of the required parabola.

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